Experiments with cellular structures

这个建议的对象是五类代数感兴趣的表示论，纽结理论和数学物理：-古典舒尔代数（与代数群GLn），-群代数的对称群，- Brauer代数，-分区代数，-最近发现的舒尔代数的Brauer代数。我们将集中在一个关键的和计算可访问的一部分，其细胞结构，乘法内的细胞subconjuncents，并对有关信息的简单模块，半单性和其他基本性质将来自这种乘法。该项目包括三个主要部分：首先，我们将开发算法和计算机程序来计算相关的结构常数，即使是大型的例子。其次，我们将进行实验，特别是通过改变代数的一些离散和连续参数。第三，我们将使用实验的输出来制定精确的结构和参数的独立性，并获得一般结果的证明起点。

The objects of this proposal are five classes of algebras of interest in representation theory, knot theory and mathematical physics:– Classical Schur algebras (associated with the algebraic group GLn),– group algebras of symmetric groups,– Brauer algebras,– partition algebras,– the recently discovered Schur algebras of Brauer algebras.We will focus on a crucial and computationally accessible part of their cellular structure, the multiplication inside cellular subquotients, and on the information about simple modules, semisimplicity and other fundamental properties to be derived from this multiplication. The project consists of three main parts: First, we will develop algorithms and computer programs to calculate the relevant structure constants, even for large examples. Secondly, we will run experiments, especially by varying some of the discrete and of the continuous parameters of the algebras. Thirdly, we will use the output of the experiments to formulate precise conjectures about structure and independence of parameters and to get starting points for proofs of general results.